Linear Algebra
2.1210 min read

Subspaces

A subspace of Rm\mathbb{R}^m is a subset that is itself a vector space — it must be closed under the operations we care about. Three conditions guarantee this: it contains the zero vector, it is closed under addition (u+v\mathbf{u} + \mathbf{v} stays inside), and it is closed under scalar multiplication (cuc\mathbf{u} stays inside).

Examples: {0}\{\mathbf{0}\} (the trivial subspace), any line through the origin, any plane through the origin, and Rm\mathbb{R}^m itself. Note: a line NOT through the origin is NOT a subspace (it doesn't contain 0\mathbf{0}).

The span of any set of vectors is always a subspace. This is why spans are so important — they are the "natural" subspaces generated by a collection of vectors.

Formal View

Definition 2.12 — Subspace
A nonempty subset V⊆RmV \subseteq \mathbb{R}^m is a subspace if: \begin{enumerate} \item 0∈V\mathbf{0} \in V \item u,v∈V⇒u+v∈V\mathbf{u}, \mathbf{v} \in V \Rightarrow \mathbf{u} + \mathbf{v} \in V (closed under addition) \item u∈V,c∈R⇒cu∈V\mathbf{u} \in V, c \in \mathbb{R} \Rightarrow c\mathbf{u} \in V (closed under scalar multiplication) \end{enumerate}

Why This Matters

Subspaces are the "natural" geometric objects in linear algebra — column spaces and null spaces are both subspaces.

  • The set of solutions to Ax=0A\mathbf{x} = \mathbf{0} (null space) is always a subspace
  • The column span Col(AA) is always a subspace
  • Signal subspaces in radar and communications engineering are linear subspaces

Quiz

Question 1

The set {(x,y)∈R2:x+y=1}\{(x, y) \in \mathbb{R}^2 : x + y = 1\} is a subspace of R2\mathbb{R}^2.

Common Mistakes

  • Thinking any flat set is a subspace — it must pass through the origin.
  • Forgetting to verify all three conditions — just containing 0\mathbf{0} is not sufficient.